美国亚利桑那大学 概率学课程 考题集

MATH464,PICKRELL,MIDTERM1REVIEW

1.Rough Overview

Chapter1:Modelling of experiments with random outcomes,probability models; conditional probability and the partition theorem;independence of events.

Chapter2:Random variables,probability mass function for a discrete RV;cat-alog of important discrete RVs;expected value for a discrete RV,the formula

E(g(X))=

x

g(x)P(X=x)

conditional expectation

E(g(X)|B)=

x

g(x)P(X=x|B)

and the partition theorem.

Chapter3:Random vectors(with emphasis on two discrete random variables); probability mass function for a sequence of discrete random variables;independence of random variables;probability mass function for the sum of two independent random variables.

1.1.Problems.1.Consider a coin such that the probability of heads is p.

(a)For the experiment offlipping the coin repeatedly until3consecutive heads or3consecutive tails are obtained,let X denote the total number of tosses.Find the probability that X=1,2,3,4,5,6.The sample space is hard to describe in this case,but this is good to think about.

(b)For the experiment offlipping the coin repeatedly until3heads(not nec-essarily consecutive)are obtained,let X denote the total number of tosses.Find the probability that X=1,2,3,4,5,6.[In this case X has a negative binomial distribution with parameters p and n=3]

2.Fix a probability space(Ω,P).Suppose that P(A)and P(B)are known.

(a)If A and B are independent events,express P(A\B)in terms of P(A)and P(B).[As always,there are multiple ways to do this;you should explain your reasoning]

(b)For general A and B,if you also know P(A∩B),find P(A∪B).

2’.Prove that

P(A∩B∩C)=P(A)P(B|A)P(C|A∩B)

3.Consider the experiment offlipping a coin,with probability of heads p on each toss,n times.

(a)Describe the underlying probability space.

(b)What is special about the case p=1

2?

4.In3.,let X i denote the Bernoulli random variable which is1iffthe ith toss is a head.

(a)What are P(X i=0),P(X i=1

2),and P(X i=1)?

(b)What are the mean and variance of X i?

1

2MATH464,PICKRELL,MIDTERM1REVIEW

(c)What are the pmf,mean and variance of S=X1+..+X n?

(d)What is the joint pmf for X1and X2?

(e)What is the pmf for X1X2?for X1X2X3?

5.Each day a weatherman makes one of three predictions:”rain”,”no rain”, or”possibility of rain”.The percentages of times he makes these predictions are 10%,75%,and15%,respectively.If the forecast is”rain”,the probability of rain is 70%.If the forecast is”no rain”,the probability it will rain is20%,if the forecast is”possibility of rain”,the probability for rain is50%.

(a)Find the percentage of days on which it rains.

(b)Suppose it did not rain yesterday.What is the probability the forecast for yesterday was for”no rain”?

6.An urn contains n balls numbered1to n(n>5).I draw three balls,one at

a time without replacement.

(a)What is the probability that the three drawn are all less than5?

(b)What is the probability the three I draw are of the form k,k+1,k+2?

(c)Let X i denote the number of the ith draw,i=1,2,3.What is P(X i=1)? What is P(X2=1|X1=2)?

7.Find E(1

1+X ),if

(a)X has a Poisson distribution

(b)X has a geometric distribution.

[We will consider e tX in class]

8.Roll two four-sided dice.Let X be the number of odd dice,Y the number of even dice,and Z the number dice showing1or2.

(a)Find the joint probability densities for(X,Y)and(X,Z).

(b)Are X and Y independent?Are X and Z independent?

9.Suppose that X and Y are independent Poisson random variables with pa-rameterλ.Find the probability mass function for X+Y.

10.Show how to calculate the expected value for the standard discrete random variables(binomial with parameters p and n,Poisson,geometric).In principle you should know how to calculate the variances as well(calculating these is more time consuming).

11.If X and Y are independent discrete random variables,and assuming all quantities are well-defined,then

var(X+Y)=var(X)+var(Y)

12.Is p(k)=1

k

,k=1,2,3,..a pmf?